Mathematical Sciences: Representations of Three-Manifold Groups

9505053卡森·卡森将研究瑟斯顿在1982年提出的三维流形的几何猜想。这一猜想的证明将完成闭3-流形的拓扑分类，深入了解3-流形的拓扑性质，并得到判定两个给定的3-流形是否拓扑等价的有效方法。卡森将探索一种非常直接的方法来解决瑟斯顿猜想；给出一个3-流形M的拓扑描述，他将试图找到与M上的几何结构密切相关的M的三角剖分。柯比将继续他的工作(通常是与保罗·梅尔文一起)在拓扑量子场论中。他将试图解释康采维奇和瓦西里耶夫不变量，使用某些余维的交集-给定的3-流形的乘积的紧凑化中的两个流形，去掉对角线。他还将继续研究Witten-Reshetikhin-Turaev不变量的公式，寻找拓扑解释和应用。由于我们生活的空间是三维的，而且我们一生都只是通过日常事务来完善我们的三维几何直觉，所以外行总是惊讶地发现，数学家回答关于更高维几何对象的问题的能力往往比回答关于三维类似对象的问题的能力更强。然而，情况在很大程度上是这样的。拓扑学中最著名的杰出猜想是在世纪之交提出的三维庞加莱猜想。另一方面，它的高维类似物已经全部解决。这个项目的研究直接针对这些臭名昭著的顽固问题。***

9505053 Casson Casson will study the Geometrization Conjecture for 3- dimensional manifolds, proposed by Thurston in 1982. A proof of this conjecture would complete the topological classification of closed 3-manifolds, provide insight into topological properties of 3-manifolds, and lead to efficient methods to determine whether two given 3-manifolds are topologically equivalent. Casson will explore a very direct approach to Thurston's conjecture; given a topological description of a 3-manifold M, he will attempt to find triangulations of M that are closely related to a geometric structure on M. Kirby will continue his work (often with Paul Melvin) in Topological Quantum Field Theory. He will try to explain the Kontsevich and Vassiliev invariants, using intersections of certain codimension-two manifolds in a compactification of products of the given 3-manifold with diagonals deleted. He will also continue to study the formulae for the Witten-Reshetikhin-Turaev invariants, looking for topological interpretations and applications. Since the space that we live in is three-dimensional, and all our lives we perfect our three-dimensional geometric intuition simply by going about our everyday business, it is always surprising to the layman to learn that mathematicians' are frequently more capable of answering questions about higher dimensional geometric objects than about the analogous objects in dimension three. This is, however, very much the case. The most famous outstanding conjecture in topology is the three-dimensional Poincare conjecture, made at the turn of the century. On the other hand, its higher dimensional analogues have all been settled. The research of this project is directly addressed to these notoriously recalcitrant problems. ***